Michael Quinn reviews the history:
In 1975, after Republican Utah state legislator M. Byron Fisher sponsored a bill to ratify the proposed Equal Rights Amendment to the U.S. Constitution, the Church’s newspaper, the Deseret News, published a typically unsigned editorial, which opposed ratification. Within days, Fisher reversed himself and voted against the bill he had sponsored, explaining, "It is my church and as a bishop [of a local congregation], I’m not going to vote against its wishes."
In May 1981, U.S. senators Jake Garn and Orrin Hatch unexpectedly found themselves in the middle of a church-state conflict. The two Republicans had supported locating an MX-missile system in Utah, until the First Presidency [the Church president and his two counselors] announced its opposition to the proposal. Both quickly and compliantly changed sides, and Hatch later omitted this controversy from the discussion of "Missiles" in his memoirs.
Mormon Democrats seem to have a better track record:
When Utah Democratic senator Frank E. Moss learned that the L.D.S. Church’s living prophet had publicly endorsed the election of Richard Nixon for the U.S. presidency, he wrote to McKay, in October 1960:
Respectfully and with deepest sorrow I ask that you rectify the unfair damage done by your endorsement of Vice President Nixon as reported in the morning press. For years we have been striving to convince Americans that our church is outside of and above politics. . . . But your endorsement of one candidate over another brings tumbling down our position of church nutrality [sic] and creates ill will outside as well as inside the church. I am heartsick.
As a chronicler of Mormon politics, I’m not aware of any L.D.S. Republican officeholder who has had the chutzpah to write such a letter—from Utah’s permanent adoption of the national parties in 1891 to the 21st century.
The Dish 